Designing the Output Amplifier

This circuit amplifies the control signal in preparation for the power output transistor. Why do we need this stage at all? Because we reduced the whole signal by one-ninth in the speed-plus-position circuit.

Schematic of non-inverting output amplifier using op-amp

This op-amp amplifies the control signal to provide plenty of "punch" to the power transistor and coil.
And it isolates the rather large load of the power transistor from the capacitor's time constant and from the resistor divider bridge.

This 741 op-amp is wired as a standard non-inverting amplifier. The gain is computed from the feedback and input resistors:
Gain = (R_f + R_i) / R_i = (370K + 1.5K) / 1.5K = 247

That's a lot of gain! Do we really need that much? I'm not sure, but it worked for me! <grin> Actually, with this much gain it acts more like a binary on-off switch than a linear amplifier. It will pretty much ensure the coil is either fully "on" or completely "off". Which is a good idea for reducing power dissipation in the final output transistor. So you probably don't need a very big heat sink for your power output transistor. You might experiment with lower values of gain to see what happens.